Character amenability of real Banach algebras

Let (a, ∥.∥) be a real banach algebra. in this paper we first introduce left and right φ-amenability of a and discuss the relation between left (right, respectively) φ-amenability and φ-amenability of a for φ ∈ △(a) [ f0g where φ ∈ △(a) is the conjugate of φ. next we show that a is left (right, respectively) φ-amenable if and only if ac is left (right, respectively) φc-amenable, where ac is a suitable complexification of a and φc ∈ △(ac) is the induced character by φ on ac. in continue, we give a hereditary property for 0-amenability of a. we also study relations between the injectivity of banach left a-modules and right φ-amenability of a. finally, we characterize the left character amenability of certain real banach algebras.